[Paper Review] Edge state dynamics along curved interfaces

Guillaume Bal, Simon Becker+4 more|arXiv (Cornell University)|Jun 1, 2021

Quantum chaos and dynamical systems4 citations

TL;DR

This paper constructs dynamical edge states for Dirac operators with curved interfaces between topologically distinct media, showing that wavepackets propagate unidirectionally and dispersion-free along curved edges for long times. Using semiclassical analysis and explicit solutions for straight edges, the authors derive a family of coherent states that follow the curved interface, with curvature-induced phase shifts and lifetime limitations quantified via geometric invariants and Berry phase effects.

ABSTRACT

We study the propagation of wavepackets along weakly curved interfaces between topologically distinct media. Our Hamiltonian is an adiabatic modulation of Dirac operators omnipresent in the topological insulators literature. Using explicit formulas for straight edges, we construct a family of solutions that propagates, for long times, unidirectionally and dispersion-free along the curved edge. We illustrate our results through various numerical simulations.

Motivation & Objective

• To develop a quantitative theory for edge state dynamics along curved interfaces in topological insulators.
• To extend the concept of ballistic, dispersion-free edge states from straight to weakly curved interfaces.
• To analyze the role of curvature in limiting the lifetime of coherent edge states.
• To identify the emergence of geometric phase effects (e.g., Berry phase) in the propagation of wavepackets along closed or curved interfaces.
• To provide a rigorous mathematical framework for the propagation of wavepackets in systems with matrix-valued symbols and repeated eigenvalues.

Proposed method

• Uses a Dirac operator Hamiltonian with a spatially varying mass term κ(x), modeling topological phase transitions.
• Constructs explicit solutions for straight edges using Gaussian initial data and exact edge state profiles.
• Applies semiclassical analysis to derive effective dynamics along curved interfaces, treating the interface as a trajectory yt governed by ∇κ(yt)⊥.
• Introduces a WKB-type ansatz with time-dependent phase and amplitude, incorporating geometric variables like θt (tangent angle) and Θt = ∫₀ᵗ ˙θₛ² ds.
• Derives transport equations for the amplitude and phase, solving them via asymptotic expansion in ε, with error bounds in L².
• Uses geometric constructions to ensure |∇κ| = 1 and ∇²κ∇κ = 0 along the interface, enabling consistent parametrization and curvature analysis.

Experimental results

Research questions

• RQ1 How do edge states behave along curved interfaces, and can they remain coherent and unidirectional despite curvature?
• RQ2 What is the role of curvature in limiting the lifetime of coherent edge states in topological insulators?
• RQ3 How do geometric invariants such as the tangent angle θt and its time derivative ˙θt affect the dynamics and phase evolution of edge states?
• RQ4 Can the propagation of wavepackets along curved interfaces be described by a semiclassical trajectory that deviates from standard Hamiltonian flow?
• RQ5 What is the origin and magnitude of the geometric phase (e.g., Berry phase) observed in numerical simulations after a full revolution along a circular interface?

Key findings

• The solution to the Dirac equation with Gaussian initial data remains approximately a Gaussian wavepacket centered at yt, a unit-speed parametrization of the interface Γ, for times t ≪ ε⁻¹/².
• The wavepacket propagates unidirectionally and dispersion-free along the curved edge, with the leading-order profile given by a Gaussian modulated by spinor components e⁻ⁱθt/2 and −eⁱθt/2.
• The lifetime of the coherent state is limited by curvature, with the error bound growing as ε¹/²⟨t⟩ and the cumulative phase shift Θt = ∫₀ᵗ ˙θₛ² ds quantifying deviation from ideal propagation.
• After one full revolution along a circular interface, the phase of the wavepacket shifts by −π, matching the theoretical prediction of a Berry phase from adiabatic evolution of the effective Hamiltonian Hθ,r.
• The curvature-induced geometric phase is captured by the integral Θt = ∫₀ᵗ ˙θₛ² ds, which appears explicitly in the error estimates and amplitude corrections.
• Numerical simulations confirm that coherent propagation persists longer for asymptotically straight interfaces (e.g., tanh-type), while circular interfaces exhibit stronger dispersion due to persistent curvature effects.